Kernelizing MSO Properties of Trees of Fixed Height, and Some Consequences

Fix an integer h>=1. In the universe of coloured trees of height at most h, we prove that for any graph decision problem defined by an MSO formula with r quantifiers, there exists a set of kernels, each of size bounded by an elementary function of r and the number of colours. This yields two noteworthy consequences. Consider any graph class G having a one-dimensional MSO interpretation in the universe of coloured trees of height h (equivalently, G is a class of shrub-depth h). First, class G admits an MSO model checking algorithm whose runtime has an elementary dependence on the formula size. Second, on G the expressive powers of FO and MSO coincide (which extends a 2012 result of Elberfeld, Grohe, and Tantau)

Topics In Multivariate Statistics

Multivariate statistics concerns the study of dependence relations among multiple variables of interest. Distinct from widely studied regression problems where one of the variables is singled out as a response, in multivariate analysis all variables are treated symmetrically and the dependency structures are examined, either for interest in its own right or for further analyses such as regressions. This thesis includes the study of three independent research problems in multivariate statistics. The first part of the thesis studies additive principal components (APCs for short), a nonlinear method useful for exploring additive relationships among a set of variables. We propose a shrinkage regularization approach for estimating APC transformations by casting the problem in the framework of reproducing kernel Hilbert spaces. To formulate the kernel APC problem, we introduce the Null Comparison Principle, a principle that ties the constraint in a multivariate problem to its criterion in a way that makes the goal of the multivariate method under study transparent. In addition to providing a detailed formulation and exposition of the kernel APC problem, we study asymptotic theory of kernel APCs. Our theory also motivates an iterative algorithm for computing kernel APCs. The second part of the thesis investigates the estimation of precision matrices in high dimensions when the data is corrupted in a cellwise manner and the uncontaminated data follows a multivariate normal distribution. It is known that in the setting of Gaussian graphical models, the conditional independence relations among variables is captured by the precision matrix of a multivariate normal distribution, and estimating the support of the precision matrix is equivalent to graphical model selection. In this work, we analyze the theoretical properties of robust estimators for precision matrices in high dimensions. The estimators we analyze are formed by plugging appropriately chosen robust covariance matrix estimators into the graphical Lasso and CLIME, two existing methods for high-dimensional precision matrix estimation. We establish error bounds for the precision matrix estimators that reveal the interplay between the dimensionality of the problem and the degree of contamination permitted in the observed distribution, and also analyze the breakdown point of both estimators. We also discuss implications of our work for Gaussian graphical model estimation in the presence of cellwise contamination. The third part of the thesis studies the problem of optimal estimation of a quadratic functional under the Gaussian two-sequence model. Quadratic functional estimation has been well studied under the Gaussian sequence model, and close connections between the problem of quadratic functional estimation and that of signal detection have been noted. Focusing on the estimation problem in the Gaussian two-sequence model, in this work we propose optimal estimators of the quadratic functional for different regimes and establish the minimax rates of convergence over a family of parameter spaces. The optimal rates exhibit interesting phase transition in this family. We also discuss the implications of our estimation results on the associated simultaneous signal detection problem

Regression on fixed-rank positive semidefinite matrices: a Riemannian approach

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semidefinite matrix. Good performance is observed on classical benchmarks

The Degrees of Freedom of Partial Least Squares Regression

The derivation of statistical properties for Partial Least Squares regression can be a challenging task. The reason is that the construction of latent components from the predictor variables also depends on the response variable. While this typically leads to good performance and interpretable models in practice, it makes the statistical analysis more involved. In this work, we study the intrinsic complexity of Partial Least Squares Regression. Our contribution is an unbiased estimate of its Degrees of Freedom. It is defined as the trace of the first derivative of the fitted values, seen as a function of the response. We establish two equivalent representations that rely on the close connection of Partial Least Squares to matrix decompositions and Krylov subspace techniques. We show that the Degrees of Freedom depend on the collinearity of the predictor variables: The lower the collinearity is, the higher the Degrees of Freedom are. In particular, they are typically higher than the naive approach that defines the Degrees of Freedom as the number of components. Further, we illustrate how the Degrees of Freedom approach can be used for the comparison of different regression methods. In the experimental section, we show that our Degrees of Freedom estimate in combination with information criteria is useful for model selection.Comment: to appear in the Journal of the American Statistical Associatio